Best regards,A Fellow Scientist

[aihaike]

Hi guys,

Einstein potential is define as

$$U_{e}=\sum_{i}\alpha_{i}(r_{i}-r0_{i})$$.

The partition function of the Hamiltonian

$$H=U_{e}+\frac{p^{2}}{2m}$$

is given by

$$Q=\left(\frac{2m_{i}}{\alpha_{i}}\left(\frac{\pi}{\beta h}\right)^{2}\right)^{\frac{3}{2}}$$

Which gives rises to the free energy

$$A=-\frac{3}{2\beta}\sum_{i=1}^{N}\ln\left[\frac{2m_{i}}{\alpha_{i}}\left(\frac{\pi}{\beta h}\right)^{2}\right]$$

Ok, now suppose we have two species in the system.
We can reformulate the partition function introducing $$\omega_{i}$$ and $$T_{\mathrm{E}_{i}}$$ for each species difine as

$$\omega_{i}=\sqrt{\frac{2\alpha_{i}}{m_{i}}}\quad\mathrm{and}\quad T_{\mathrm{E}_{i}}=\frac{h\omega_{i}}{2\pi k}$$

And it comes

$$A=3N_{1}kT\ln\left(\frac{T_{\mathrm{E}_{1}}}{T}\right)+3N_{2}kT\ln\left(\frac{T_{\mathrm{E}_{2}}}{T}\right)$$

where $$N_{1}$$ and $$N_{2}$$ are the number of atom of each species.

We also can take the quantum version of the partition function defines as

$$Q={\left(\frac{\exp\left(-\frac{T_{\mathrm{E}_{1}}}{2T}\right)}{1-\exp\left(-\frac{T_{\mathrm{E_{1}}}}{T}\right)}\right)^{3N_{1}}} {\left(\frac{\exp\left(-\frac{T_{\mathrm{E}_{2}}}{2T}\right)}{1-\exp\left(-\frac{T_{\mathrm{E_{2}}}}{T}\right)}\right)^{3N_{2}}}$$

Now comes my question.
On one hand it seems to me that the Einstein temperature $$T_{\mathrm{E}$$ is a characteristic is the system but it seems also depends on the system temperature.

I'm working on silica (SiO2) and I use to compute for each species $$\alpha$$ from the mean square displacement (u) calculated from a NVT monte carlo simulation (with the initial structure from the average coordinates of NPT simulation) using a standard 2-body potential with the formula

$$\alpha=\frac{3KT}{2u}$$

The I calculate the corresponding frequency and Einstein temperature

$$\omega=\sqrt{\frac{2\alpha}{m}}\quad\mathrm{and}\quad T_{\mathrm{E}}=\frac{h\omega}{2\pi k}$$

And the $$T_{\mathrm{E}$$ I get does not correspond to my apply temperature.
May be I should call it vibrational temperature instead of Einstein temperature.

I'm very confused, if we want to calculate the free energy of a crystal based on this model, what value of $$T_{\mathrm{E}$$ shoud we take? Is there table somewhere?
I also wonder if $$T_{\mathrm{E}$$ and $$\alpha$$ are really related actually.
Any comments are welcome.
Thanks in advance,

Eric.

 

[eljose79]

Dear Eric,

Thank you for sharing your thoughts on the Einstein potential and its relation to the partition function and free energy. It is clear that you have put a lot of effort into understanding this concept and its application to your research on silica.

To answer your question, the value of T_{\mathrm{E} that should be used in the calculation of the free energy depends on the specific system and conditions being studied. In general, the Einstein temperature can be considered a characteristic temperature of the system, but it can also vary depending on the temperature at which the system is being studied.

In your case, as you have mentioned, the value of T_{\mathrm{E} calculated from the mean square displacement and the corresponding frequency may not correspond to the applied temperature. This could be due to various factors, such as the limitations of the potential used in your simulations or the specific conditions of your system. It is also possible that the vibrational temperature, as you mentioned, may be a more appropriate term to use in this context.

In terms of finding a table for T_{\mathrm{E} and \alpha, it is important to note that these values can vary significantly depending on the specific system and conditions being studied. Therefore, it is not common to have a general table for these values. It would be more appropriate to calculate them specifically for your system using the methods you have mentioned.

In summary, the value of T_{\mathrm{E} to be used in the calculation of free energy can vary and should be determined based on the specific system and conditions being studied. It is also important to consider the limitations and uncertainties associated with the calculation of T_{\mathrm{E} and \alpha. I hope this helps clarify your confusion. Best of luck with your research!

 

[charng]

Hi Eric,

Thank you for sharing your thoughts and questions about the Einstein solid model. It seems like you have a good understanding of the equations and concepts involved, but are struggling with some practical applications and interpretations. Let me try to address your concerns and provide some clarification.

Firstly, the Einstein potential and partition function are important tools in statistical mechanics for understanding the behavior of a system of particles, such as atoms in a solid. In this model, the potential energy is described by the sum of the displacements of each particle from its equilibrium position, and the partition function takes into account the energy and momentum of each particle. These equations are useful for calculating thermodynamic properties, such as the free energy, of the system.

As for your question about the Einstein temperature, it is indeed a characteristic of the system but it does depend on the system temperature. This is because the Einstein temperature is related to the average kinetic energy of the particles, which is directly influenced by the temperature of the system. Therefore, the Einstein temperature can be seen as a measure of the temperature at which the particles in the system vibrate with the most energy.

In your work with silica, it is important to note that the Einstein temperature and vibrational temperature are not the same thing. The Einstein temperature is a theoretical concept, while the vibrational temperature is a practical measurement based on the mean square displacement of particles. It is possible that the values you are calculating for the Einstein temperature do not match your applied temperature because of the limitations of the model and the approximations made in your calculations. However, these values can still be useful in understanding the behavior of your system.

Finally, in terms of calculating the free energy of a crystal based on this model, it is important to use the appropriate value of the Einstein temperature for your system. This value can be determined experimentally or theoretically, but it is not a fixed value and can vary depending on the system and its conditions. I am not aware of a table specifically for Einstein temperatures, but there may be tables or databases for related properties that could be helpful in your research.

I hope this helps to clarify some of your questions and concerns. Keep exploring and questioning the Einstein solid model, as it is a valuable tool in understanding the behavior of solids. Best of luck in your research!